Implication (2) \(\Rightarrow \)  (3) is an immediate consequence of Theorem REF . If \(\beta < \alpha \)  then we define the operator of restriction:
\(T:f\mapsto f|_{[1,\beta ]}\) . If \(\alpha \le \beta \) , then the assumption of (3) says that there is an isomorphism between \(C_{p}([1,\alpha ])\)  and \(C_{p}([1,\beta ])\) 
which trivially implies (3) \(\Rightarrow \)  (1). Now we prove (1) \(\Rightarrow \)  (2).
It is probably a folklore that (1) implies that \(Y\)  has to be a countable compact space. For instance, one can argue as follows.
First, \(Y\)  is a metrizable compact space by [1]}. Second, \(Y\)  is scattered by [2]}.
Every scattered metrizable compact space is homeomorphic to \([1,\beta ]\)  for some countable ordinal \(\beta \) . If \(\beta < \alpha \)  there is nothing to prove.
So, let us assume that \(\alpha \le \beta \) .
Applying the Closed Graph Theorem we consider \(T\)  as a linear bounded operator from the Banach space \(C([1,\alpha ])\)  onto the Banach space \(C([1,\beta ])\) .
Note that in a particular case \(\alpha = \omega \)  we have that \(C([1,\alpha ]) \approx c_{0}\) , hence \(C([1,\beta ])\)  is a quotient of \(c_0\) ,
and we can apply the theorem of W.B. Johnson and M. Zippin [3]} to deduce that the space \(C([1,\beta ])\)  is isomorphic to \(c_0\) . By Theorem REF  we conclude that in
this particular case \(\beta < \omega ^{\omega }\) .
In a general case of arbitrary countable ordinal \(\alpha \)  we will apply more elaborate results of functional analysis.
Recall that the Szlenk index of a Banach space \(E\) , denoted \({\rm Sz}(E)\) , is an ordinal number, which is invariant under linear isomorphisms.
For the definition of \({\rm Sz}(E)\)  and its basic properties we refer the reader to the survey papers [4]}, [5]}.
The key tool is the following precise result of C. Samuel, which is in turn based on a deep result of D. Alspach and Y. Benyamini [6]},
(see [4]}, [5]}).
Fact A. For any \(0 \le \gamma <\omega _1\)
\({\rm Sz}(C([1,\omega ^{\omega ^{\gamma }}])) = \omega ^{\gamma +1}.\) 
We need also
Fact B. [5]} Let \(E_1\)  and \(E_2\)  be given Banach spaces with norm-separable duals.
Assume that \(E_2\)  is isomorphic to a subspace of a quotient space of \(E_1\) .
Then \({\rm Sz}(E_2) \le {\rm Sz}(E_1)\) .
In order to finish the proof of (1) \(\Rightarrow \)  (2) suppose the contrary: \(\beta \ge \alpha ^{\omega }\) .
Then by Fact A, \({\rm Sz}(C([1,\beta ])) > {\rm Sz}(C([1,\alpha ]))\)  which contradicts Fact B.
